Moment estimates implied by modified log-Sobolev inequalities

TL;DR

This paper explores a broad class of modified log-Sobolev inequalities, establishing their implications for moment estimates, concentration inequalities, and non-Euclidean gradient norms, extending classical results.

Contribution

It generalizes modified log-Sobolev inequalities and derives new $L^p$-Sobolev inequalities with non-Euclidean norms, leading to enhanced concentration results.

Findings

Modified inequalities imply $L^p$-Sobolev inequalities with non-Euclidean norms

Derived concentration inequalities for smooth functions and set enlargements

Established a two-level concentration for functions with bounded Hessian

Abstract

We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a method of Aida and Stroock for the classical logarithmic Sobolev inequality, we prove that if a measure on $\mathbb{R}^n$ satisfies a modified logarithmic Sobolev inequality then it satisfies a family of $L^p$-Sobolev-type inequalities with non-Euclidean norms of gradients (and dimension-independent constants). The latter are shown to yield various concentration-type estimates for deviations of smooth (not necessarily Lipschitz) functions and measures of enlargements of sets corresponding to non-Euclidean norms. We also prove a two-level concentration result for functions of bounded Hessian and measures satisfying the classical logarithmic Sobolev inequality.